Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from specific cases to general rules, i.e., there is epistemic uncertainty. However, the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.

The first premise states that all objects classified as "men" have the attribute "mortal". The second premise states that "Socrates" is classified as a "man" – a member of the set "men". The conclusion then states that "Socrates" must be "mortal" because he inherits this attribute from his classification as a "man".

The law of detachment (also known as affirming the antecedent and Modus ponens) is the first form of deductive reasoning. A single conditional statement is made, and a hypothesis (P) is stated. The conclusion (Q) is then deduced from the statement and the hypothesis. The most basic form is listed below:

P → Q (conditional statement)

P (hypothesis stated)

Q (conclusion deduced)

In deductive reasoning, we can conclude Q from P by using the law of detachment.[2] However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no definitive conclusion.

The following is an example of an argument using the law of detachment in the form of an if-then statement:

If an angle satisfies 90° < A < 180°, then A is an obtuse angle.

A = 120°.

A is an obtuse angle.

Since the measurement of angle A is greater than 90° and less than 180°, we can deduce that A is an obtuse angle. If however, we are given the conclusion that A is an obtuse angle we cannot deduce the premise that A = 120°.

The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:

P → Q

Q → R

Therefore, P → R.

The following is an example:

If Larry is sick, then he will be absent.

If Larry is absent, then he will miss his classwork.

Therefore, if Larry is sick, then he will miss his classwork.

We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the second statement. We also allow that this could be a false statement. This is an example of the transitive property in mathematics. The transitive property is sometimes phrased in this form:

An argument is “valid” if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be “valid” even if one or more of its premises are false.

An argument is “sound” if it is valid and the premises are true.

It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form.

The following is an example of an argument that is “valid”, but not “sound”:

Everyone who eats carrots is a quarterback.

John eats carrots.

Therefore, John is a quarterback.

The example’s first premise is false – there are people who eat carrots who are not quarterbacks – but the conclusion would necessarily be true, if the premises were true. In other words, it is impossible for the premises to be true and the conclusion false. Therefore, the argument is “valid”, but not “sound”. False generalizations – such as “Everyone who eats carrots is a quarterback” – are often used to make unsound arguments. The fact that there are some people who eat carrots but are not quarterbacks proves the flaw of the argument.

Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is “valid”, it is possible for the conclusion to be false (determined to be false with a counterexample or other means).